Foods and Raw Materials Foods and Raw Materials Foods and Raw Materials 2308-4057 2310-9599 52667 10.21603/2308-4057-2022-2-542 Research Article Research Article Research Article Thermodynamic factor and vacuum crystallization Thermodynamic factor and vacuum crystallization https://orcid.org/0000-0001-6203-1783 Semenov Evgeniy V. Semenov Evgeniy V. https://orcid.org/0000-0002-0262-8841 Slavyanskiy Anatoliy A. Slavyanskiy Anatoliy A. slavyanskiyaa@mgutm.ru https://orcid.org/0000-0002-8724-3368 Mitroshina Daria P. Mitroshina Daria P. d.mitroshina@mgutm.ru https://orcid.org/0000-0003-4936-7498 Lebedeva Natalya N. Lebedeva Natalya N. K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Россия K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Russian Federation K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Россия K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Russian Federation K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Россия K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Russian Federation K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Россия K.G. Razumovsky Moscow State University of Technologies and Management (the First Cossack University) Moscow Russian Federation 23 09 2022 23 09 2022 10 2 304 309 22 03 2022 01 07 2022 https://jfrm.ru/en/issues/20341/20540/

Sucrose crystallization depends on various thermal phenomena, which makes them an important scientific issue for the sugar industry. However, the rationale and theory of sucrose crystallization still remain understudied. Among the least described problems is the effect of time and temperature on the condensation rate of sucrose molecules on crystallization nuclei in a supersaturated sugar solution. This article introduces a physical and mathematical heat transfer model for this process, as well as its numerical analysis. The research featured a supersaturated sugar solution during sucrose crystallization and focused on the condensation of sucrose molecules on crystallization nuclei. The study involved the method of physical and mathematical modeling of molecular mass transfer, which was subjected to a numerical analysis. While crystallizing in a vacuum boiling pan, a metastable solution went through an exothermal reaction. In a supersaturated solution, this reaction triggered a transient crystallization of solid phase molecules and a thermal release from the crystallization nuclei into the liquid phase. This exogenous heat reached 39.24 kJ/kg and affected the mass transfer kinetics. As a result, the temperature rose sharply from 80 to 86 °C. The research revealed the effect of temperature and time on the condensation of solids dissolved during crystalline sugar production. The model involved the endogenous heat factor. The numerical experiment proved that the model reflected the actual process of sucrose crystallization. The obtained correlations can solve a number of problems that the modern sugar industry faces.

Sucrose crystallization depends on various thermal phenomena, which makes them an important scientific issue for the sugar industry. However, the rationale and theory of sucrose crystallization still remain understudied. Among the least described problems is the effect of time and temperature on the condensation rate of sucrose molecules on crystallization nuclei in a supersaturated sugar solution. This article introduces a physical and mathematical heat transfer model for this process, as well as its numerical analysis. The research featured a supersaturated sugar solution during sucrose crystallization and focused on the condensation of sucrose molecules on crystallization nuclei. The study involved the method of physical and mathematical modeling of molecular mass transfer, which was subjected to a numerical analysis. While crystallizing in a vacuum boiling pan, a metastable solution went through an exothermal reaction. In a supersaturated solution, this reaction triggered a transient crystallization of solid phase molecules and a thermal release from the crystallization nuclei into the liquid phase. This exogenous heat reached 39.24 kJ/kg and affected the mass transfer kinetics. As a result, the temperature rose sharply from 80 to 86 °C. The research revealed the effect of temperature and time on the condensation of solids dissolved during crystalline sugar production. The model involved the endogenous heat factor. The numerical experiment proved that the model reflected the actual process of sucrose crystallization. The obtained correlations can solve a number of problems that the modern sugar industry faces.

 Vacuum boiling pan sucrose phase metastable solution heat dissolution condensation crystallization Vacuum boiling pan sucrose phase metastable solution heat dissolution condensation crystallization

INTRODUCTIONVacuum boiling pans are an essential componentof sugar and starch production. A vacuum pan is acrystallizer filled with a liquid solution of sucrose, salts,or other substances.A metastable liquid solution behaves like ahomogeneous liquid. If it is oversaturated, a thinsuspension or a solid phase introduced into thecrystallization nuclei can trigger a rapid and powerfulthermal reaction. This reaction turns the homogeneoussolution into a heterogeneous liquid system calledmassecuite.The thermal release during crystal formation iscaused by two factors. On the one hand, the force ofattraction accelerates the flow of sucrose moleculesto the crystallization nuclei. On the other hand, whenthe molecule clusters stop on the surface of thecrystallization nucleus, the accumulated kineticenergy is spent on embedding the molecules into thecrystal lattice, as well as on internal energy. As aresult, molecules get accumulated on the crystallizationnucleus, and this process is known as crystallization ofsucrose in a vacuum pan.In the sugar industry, energy production relies on allphysical forms of thermal energy of water, be it liquidor vaporized. Thermal equipment turns water into steam,which acts as the main heat generator to obtain sugar orsugar products. After that, the steam serves as a heaterand evaporates moisture from another heterogeneousliquid system, e.g., beet juice. The steam can also go intoa new physical state: it settles on the cooled solid walls ofthe equipment, turns into a liquid, and releases the heat.305Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309This phenomenon illustrates the law of energyconservation. Water molecules move in this gaseousmedium and settle down on the equipment walls. Asvapor transforms into liquid, it releases thermal energy,which is a powerful and efficient reaction. As a result,the temperature inside the environment rises, whichmakes steam the main source of thermal energy in sugarproduction.The same law of energy conservation is responsiblefor crystallization, which occurs in a supersaturatedsugar solution when the distance between thecrystallization nuclei becomes small enough to triggerthe forces of attraction between sucrose molecules.Hence, crystallization happens when sucrose moleculesconcentrate on the surface of the crystallization nuclei.A lot of studies concentrate on the scientific andtechnical issues of metastable and supersaturatedsolutions because these phenomena are crucial for sugarproduction technology [1–19].For instance, Saifutdinov et al. focused on the effectof various organic solvents on the molar changes inthe Gibbs energy, enthalpy, and entropy during adsorption[1]. They established the role of intermolecularinteractions in the solution and at the phase boundary.In another article, Saifutdinov et al. reported theadsorption thermodynamics for some 1,3,4-oxadiazolesand 1,2,4,5-tetrazines from water-acetonitrile and watermethanolsolutions on the surface of porous graphitizedcarbon at 313–333 K [2]. The absolute values of thechange in the Gibbs energy and enthalpy increasedduring the adsorption from water-organic solutions asthe surface area of adsorbate molecules became larger,the absolute values of the change in entropy decreased,and the Van der Waals volume of molecules increased.Makhmudov et al. calculated the thermodynamicparameters for the phenol and sulfonol sorption fromwastewater on activated carbon and anion exchanger [3].Sagitova et al. described the sorption of cobaltions by native and modified organic pharmacophoresof pectins [4]. They determined the effect of acidity,temperature, and solution/sorbent module on thedistribution of cobalt ions in the heterophase systemof polysaccharide sorbent and aqueous solution. Thisresearch also revealed the effect of various biosorbentson the thermodynamics of cobalt ions.Sharma et al. used the method of isothermalmicrocalorimetry to determine the dilution enthalpy offluorosiloxane rubber and polychloroprene solutions invarious organic liquids [5]. The dissolution processesof polychloroprene were accompanied by exothermicprocesses, while those of fluorosiloxane rubber – byendothermic ones.Sayfutdinov and Buryak applied liquid chromatographyto study the adsorption of isomericdipyridyls and their derivatives from aqueousacetonitrile, aqueous methanol, and aqueous isopropanolsolutions on a graphite-like carbon [6].Fedoseeva and Fedoseev proved that size changesthe state and physicochemical properties of dispersedsystems in small (nano-, pico-, femtoliter) volumes [7].The scientists used digital optical microscopy tointerpret the concepts of chemical thermodynamics.Their experiments established the effect of suchgeometric parameters as radius and contact angle onthe kinetics of phase and chemical transformations. Theresearch featured polydisperse accumulations of dropletsin organic and water-organic mixes that interacted withvolatile gaseous reagents.Other publications reported on the kinetics, mechanism,and heat of crystallization processes [8–15].Some of them [8–10] focused on phase thermal effects inthe sugar industry based on the laws of thermodynamicsand the Gibbs theory.Jamali et al. studied such independent kineticfactors as thermodynamics and sucrose crystal transferthat occur in an aqueous sugar solution duringcrystallization [16]. They used high-precision tools andscaling to prove that the experimental results confirmedthe precalculated fluid densities, thermodynamic factors,shear viscosity, self-diffusion coefficients, and the Fickdiffusion coefficients.Li et al. described a modern view on crystalnucleation [17]. Traditional physical organic chemistryalways combined kinetics and thermodynamics tostudy crystallization. The authors studied sucrose andp-aminobenzoic acid to show how solution chemistry,crystallography, and kinetics complement each other toprovide a complete picture of all nucleation processes.Kumagai proved the effect of the water sorptionisotherm on the interaction of water and solids in foodproducts [18]. In thermodynamics, the Gibbs free energy(ΔGs) describes the interaction of a solid substance andwater. Therefore, the plasticizing effect of water on foodproducts can be evaluated by applying the Gibbs freeenergy.Ebrahimi et al. studied a mix of 1-butanol + waterwith or without sugars and their effect on clouding [19].This experiment established that 1-butanol + watersolution fortified with sucrose or alcohol reducedclouding.These publications give a thorough account of phasetransition of liquid to vapor and back, but they providea poor quantitative assessment of the heat released orabsorbed in each case.The present paper introduces the thermal problemof heat propagation in the intercrystal solution volumeadjacent to crystallization nuclei (instantaneous heatsource).STUDY OBJECTS AND METHODSThe research featured a supersaturated sugarsolution in a vacuum boiling pan under the conditions ofindustrial sugar production.The methods included physical and mathematicalmodeling of heat and mass transfer in heterogeneousliquid systems.Modeling. Heat transfer in a vacuum pan is adifficult task for physical and mathematical modeling,306Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309while its numerical calculation provides a scheme thatreflects the actual process [13].The modeling relied on the assumption thatcrystallization nuclei are uniformly distributed in thevacuum pan. Therefore, the calculations relied on thespherical symmetry of the liquid + solid mix relative tocenter O in the region of 0 < r < R, where r is the radiusof the model sphere and R is the average radial distancebetween the spheres (Fig. 1).The boundary value problem was based on the theoryof thermal conductivity for an isolated model particleof sucrose near the crystallization nucleus. A certainvolume of intercrystalline solution was represented asa spherical region with radius R and center point O atsaturation temperature Тs. The volume included a modelsucrose particle represented as a sphere with radiusr = r1 and center O. The initial instantaneous heat sourcedistributed over spherical surface r = r1 (Fig. 1) withforce Q1 (J). Heat exchange occurred in accordance withthe boundary condition of the third kind between spheresurface r = R and its environment. The task was to findthe temperature field in the region of 0 < r < R and theaverage temperature of the medium over time.The heat transfer equation looks as follows:where Т(r,τ) is the temperature, K; τ is the time, s; а isthe thermal diffusion coefficient, m2/s.The initial data include:wheretemperature difference between sphere surface r = r1and the environment, K; Qsp is the specific heatof crystallization, J/kg; с0 is the heat capacity of thesolution, J/(kg⋅ K).Boundary conditions:where Т0 and Т1 are the initial temperature (К) of theenvironment (massecuite) and the temperature onsphere surface r = R, m, respectively; Н = α/λ, α isthe thermal diffusion coefficient, Vt/(m2⋅K); λ is thethermal conduction coefficient, Vt/(m⋅K).If we introduce the following substitution(6)the boundary problem (1)–(5) looks as follows:(7)(8)(9)(10)where t(r,τ) is the reduced temperature, Δt = Т0 – Т1, andδТ is defined according to (3).Boundary problems (7)–(10) are based on thefollowing correlation [20]:(11)– sucrose crystal volume, m3;and t(r,0) as in (8), K; μ1 and μ2, are the roots of thecharacteristic equation,(12)Вi = αh/λ – Biot number (thermal), Fo = ατ/R2 –Fourier number [20].According to (11),(13)where Аn is the table coefficients [20].Formula (6) provides the following solution for(7)–(10):(14)where t(r,τ) is the calculated according to (13).Mean temperature 0 < r < R is calculated as follows:(15)where function T(r,τ) under the integral depends oncorrelation (14).Figure 1 Heat and mass transfer for sucrose crystallization ina vacuum boiling panSucrose crystalIntercrystal solutianIntercrystallinesolution(1)(2)(3)(4)(5)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) 3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ 𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 ⁄ 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟< 𝑅𝑅𝑅𝑅)𝑟𝑟𝑟𝑟, 0) 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑉𝑉𝑉𝑉 = 4𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟13⁄3𝑡𝑡𝑡𝑡g𝜇𝜇𝜇𝜇 = −𝜇𝜇𝜇𝜇⁄(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)𝑉𝑉𝑉𝑉 = 4𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟13⁄3𝑡𝑡𝑡𝑡g𝜇𝜇𝜇𝜇 = −𝜇𝜇𝜇𝜇⁄(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) − 𝑇𝑇𝑇𝑇1𝜕𝜕𝜕𝜕[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝜏𝜏𝜏𝜏= 𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕2[𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)]𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟2 (𝜏𝜏𝜏𝜏 > 0, 0 < 𝑟𝑟𝑟𝑟 < 𝑅𝑅𝑅𝑅)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0) = 𝑛𝑛𝑛𝑛 􀵜(Δ𝑡𝑡𝑡𝑡 + 𝛿𝛿𝛿𝛿𝑇𝑇𝑇𝑇)at 𝑟𝑟𝑟𝑟 ≤ 𝑟𝑟𝑟𝑟1Δ𝑡𝑡𝑡𝑡 at 𝑟𝑟𝑟𝑟 > 𝑟𝑟𝑟𝑟1𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟= 0, 𝑡𝑡𝑡𝑡(0, 𝜏𝜏𝜏𝜏) ≠ ∞, at 𝜏𝜏𝜏𝜏 > 0𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏)𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟+ 𝐻𝐻𝐻𝐻𝑡𝑡𝑡𝑡(𝑅𝑅𝑅𝑅, 𝜏𝜏𝜏𝜏) = 0𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 − sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 cos𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟1∙ sin𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟⁄𝑅𝑅𝑅𝑅𝑟𝑟𝑟𝑟∙ exp(−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0),𝑏𝑏𝑏𝑏 = 𝑉𝑉𝑉𝑉 ∙ 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 0)𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) =𝑏𝑏𝑏𝑏4𝜋𝜋𝜋𝜋𝑅𝑅𝑅𝑅∙ 􀷍1𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟1∞𝑛𝑛𝑛𝑛=1∙[(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 1)2 + 𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2]1⁄2 𝐴𝐴𝐴𝐴𝑛𝑛𝑛𝑛𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵∙ sin (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟1 𝑅𝑅𝑅𝑅) ∙ sin ⁄ (𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅) ∙ exp ⁄ (−𝜇𝜇𝜇𝜇𝑛𝑛𝑛𝑛 2𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏) = 𝑇𝑇𝑇𝑇1 + 𝑡𝑡𝑡𝑡(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚(𝜏𝜏𝜏𝜏) =3𝑅𝑅𝑅𝑅3 􀶱 𝑇𝑇𝑇𝑇(𝑟𝑟𝑟𝑟, 𝜏𝜏𝜏𝜏)𝑟𝑟𝑟𝑟2𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑅𝑅𝑅𝑅0307Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309The temperature and the mean temperature inthe vacuum pan depend on the processing time andare calculated based on correlations (14) and (15).As follows from the assumption about the uniformdistribution of the crystallization nuclei, the calculatedthermal characteristics for the selected elementaryvolume with radius R are also valid for the entire volumeof the vacuum pan.RESULTS AND DISCUSSIONThe initial data included: crystal radiusr1 = 1×10–5 and 2×10–5 m; volume concentration с = 40and 50% (с = 0.4, 0.5); density of intercrystallinesolution (massecuite) ρ = 1450 kg/m3 [10]; thermalconduction and diffusion coefficient (for water at80°С), respectively, λ = 0.56 Vt/(m⋅°С), с0 = 1250 J/(rg⋅K),heat transfer coefficient α = 240 Vt/(m2⋅°C) [21].The resulting thermal diffusion coefficient isа = λ/(с0⋅ρ) = 3.09×10–7 m2/s. The equivalent radius ofelementary volume was calculated as follows:R = r1⋅с–1/3 (16)Biot number Bi = α⋅r1/(λ⋅с1/3).The specific heat of sucrose crystallization wasas in [13]: Qsp = 13.42 kJ/mol (39.24 kJ/kg).The numerical simulation was based on MATHCADsoftware.Sum (13) was calculated based on (12)–(16) withthe same four additive components, while theparameters of Аn and μn in (13) were based on the tablespublished in [20].Temperatures Т0 and Т1 were 80°С all the time,which means that ΔТ (9) = 0.Figures 2 and 3 show the calculation results atthe accepted values of the thermal process: volumeconcentration c of the solid phase in the solution,time τ, and temperature Т on surface r1 formodel sucrose particle and mean massecuitetemperature Т.Figures 2 and 3 show that the heat transfer intothe sugar solution during crystallization of the modelsucrose particle proceeded very quickly and took somethousandths of a second. That was why the thermalregime in the intercrystalline solution stabilized soquickly.Figures 2 and 3 also demonstrate the samegradual exponential decrease in temperature, whichis typical for heat transfer problems. If particlesdiffered in radius by a factor of two, smaller particleswith a larger specific surface area and a greaterheat transfer cooled faster than particles with alarger radius. For curves 1 and 2, the temperaturerise rate of the particles with radius r1 = 1×10–5 mexceeded curves 3 and 4 for particles with a radiustwice as large. The accumulation and releaseof heat for crystals with radius r1 = 2×10–5 mwas eight times bigger than those for crystals with aradius two times smaller. Figure 3 clearly demonstratesthat curves 3 and 4 are much higher thancurves 1 and 2.CONCLUSIONSThe equation of non-stationary Fourier diffusionwith initial and boundary conditions of the third kindwas applied to calculate the endogenous heat releasedinto the solution during the condensation of sucrosemolecules on a spherical particle of a sucrose crystal ina supersaturated sugar solution.The numerical study involved conditions close tothe actual sucrose crystallization process in a vacuumboiling pan. It revealed an increase in temperature as aFigure 3 Correlation of mean massecuite temperatureТ with volume concentration с of the solid phase in thesolution and crystallization time τ (r1 = 1×10–5 m:1 – с = 40%, 2 – с = 50%; r1 = 2×10–5 m: 3 – с = 40%,4 – с = 50%)Figure 2 Correlation of temperature Т on surface r1 of themodel sucrose particle with volume concentration с of thesolid phase in the solution and crystallization time τ(r1 = 1×10–5 m: 1 – с = 40%, 2 – с = 50%; r1 = 2×10–5 m:3 – с = 40%, 4 – с = 50%)8081828384858687881 2 3 4 5τ×10–3, sT,°C1 2 3 48081828384858687881 2 3 4 5τ×10–3, sT,°C1 2 3 48182838485860.5 1.0 1.5T,°Cτ×10–3, s1 2 3 48081828384858687881 2 3 4 5T,°C1 2 3 48182838485860.5 1.0 1.5T,°Cτ×10–3, s1 2 3 4308Semenov E.V. et al. Foods and Raw Materials. 2022;10(2):304–309result of the phase transition from 80 to 86°С in 2×10–3 s,which means the process was almost instantaneous.The calculations were confirmed in practice.The results can facilitate calculating the effect oftemperature on massecuite viscosity, wash watertemperature, and other characteristics of massecuitevacuum processing in the sugar and starchindustries.CONTRIBUTIONE.V. Semenov and A.A. Slavyanskiy supervised theproject. D.P. Mitroshina and N.N. Lebedeva performedthe experiments.CONFLICT OF INTERESTThe authors declare that there is no conflict ofinterests regarding the publication of this article.

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